# What is the derivative of f(t) = (t^2-lnt, t^2-sint ) ?

Feb 10, 2016

$f ' \left(t\right) = \frac{2 {t}^{2} - t \cos t}{2 {t}^{2} - 1}$

#### Explanation:

We know that

$x \left(t\right) = {t}^{2} - \ln t$

$y \left(t\right) = {t}^{2} - \sin t$

Differentiate each of these with respect to $t$:

$x ' \left(t\right) = 2 t - \frac{1}{t}$

$y ' \left(t\right) = 2 t - \cos t$

The derivative of the parametric function is equal to

$f ' \left(t\right) = \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y ' \left(t\right)}{x ' \left(t\right)} = \frac{2 t - \cos t}{2 t - \frac{1}{t}}$

Multiply the function by $\frac{t}{t}$ for a simplification of:

$f ' \left(t\right) = \frac{2 {t}^{2} - t \cos t}{2 {t}^{2} - 1}$